However I am not convinced that you are right about the second question.
Imagine there is only one dot, and no guest. Nobody concludes anything and so nobody dies. Hence the induction fails.
Jeroen Boschma wrote:
> Hi, > > The first evening the logicians sit down in a circle and look at > eachother. They all > focus on the number of foreheads they see WITHOUT a blue dot. On that > first evening everyone thinks: if I see 49 foreheads without a blue dot, > then I must have the only blue dot and I will commit suicide. But on > that first evening nobody commits suicide, so the overall conclusion is: > there are less then 49 foreheads whthout a blue dot (or more than 1 blue > dots). > Second evening. The reasoning is as follows: there are more than 1 blue > dots. Let's assume for this evening there are two blue dots. If I see > only one other blue dot, then I must have the other blue dot and the two > of us can commit suicide. But again, nothing happens because everybody > sees much more blue dots and the overall conclusion is that there are > more than two blue dots. > For the third evening, everybody knows that there are more than 2 blue > dots and will make their decision based on the assumption that there are > exactly 3 blue dots (if I see exactly two other dots, then I have the > third one and the three of us commit suicide). > This goes on and on, evening after evening, and on the last evening the > logicians conclude that there are more than 49, thus exactly 50, blue > dots. They party all night long, have lots of drink and women and at > dawn they commit suicide. > > Sad ending? Not at all, because as logicians they really suck. Even a > normal person would decide that if everybody has a blue dot, there's > nothing to shame about. Just a question of ... logic? > > You really don't need a guest on your isle to tell you there is at least > one blue dot. There may be a need for in other versions of this riddle > (there are many I guess), but not in this one. The guest does not supply > any information that each logician didn't know already. Or maybe I suck > as a logician, which is not that bad because I'm not one. > > Greetz, > > Jeroen > > Bill Hart wrote: > > > > For those who are interested, here is a brainteaser that took me more than > > 15 mins to solve (apologies to those who don't think it belongs here, and > > to the person I pinched it from) > > > > On the Isle of Reason live exactly 50 logicians. There are no other > > inhabitants. Each day they all meet to discuss riddles and life. However > > there is one topic they do not discuss. On each of their foreheads is a > > blue dot. However it is so shameful to have a blue dot, that knowing this > > fact for a certainty, will certainly result in the bearers' suicide the > > next morning. For this reason blue dots and related things are NEVER > > discussed. Fortunately there are no mirrors or reflections on the island > > and so everyone is blissfully unaware of the blue dot they have, but can > > see the other 49 blue dots....until, one day a visitor to the island comes > > and knowing something of the strange problems of the island thinks he will > > be tactful and announces "I happen to have noticed that at least one of you > > has a blue dot on his forehead". He leaves the island unaware of what > > follows. Precisely fifty nights pass and on the next morning all fifty > > logicians commit suicide. Why? > > > > Note: there is sufficient information in the riddle, it is not a play on > > words and can be solved by pure logic. The fact that they discuss riddles > > is irrelevant. They could come together each day without a word being > > spoken! > > > > Finally, for those who have heard the riddle, answer this. Why does someone > > need to come to the island and say that there is a blue dot? Surely each of > > them can see 49 other dots and knows that there is a blue dot? The answer > > must be as tight as the answer to the riddle! > > > > Bill.